Contribution to the modelling of chocolate tempering process

Contribution to the modelling of chocolate tempering

Proceedings of European Congress of Chemical Engineering (ECCE-6)

Copenhagen, 16-20 September 2007

Contribution to the modelling of chocolate tempering

process

F. Debaste,a Y. Kegelaers,

b H. Ben Hamor,

a V. Halloin

a

aChemical engineering department, Université Libre de Bruxelles Av. Franklin Roosevelt, 50 CP

165/67 B-1050 Brussels, Belgium bPuratos Group, Industrialaan 25, B-1702 Groot-Bijgaarden, Belgium

Abstract

The tempering of chocolate, i.e. the process of crystallization of the desired morph; is a key step in its manufacturing by professional chocolate makers. In this work, model of a tempering process based on seeding with solid chocolate grains is developed to enhance understanding and control of the system. This model aims to predict temperature field during melting and crystallization of the product. Therefore a mechanical stirrer is designed to simulate the manual mixing. Resulting flow field is modeled using CFD. Based on simulation results, the heat transfer problem is simplified using an effective thermal conductivity. The parameter value is fitted on experimental results. The heat conduction equation obtained is solved using Femlab. Melting of chocolate particles serving as seeds is added to the model using a sink term having the form of a kinetic reaction whose parameters are identified from an adiabatic melting experiment. The resulting model gives an accurate prediction of the cooling rate and the temperature field within the melted chocolate seeded with small solid grains. Keywords: Chocolate tempering, Heat transfer, Computational fluid dynamics

1. Introduction

Tempering process refers to a controlled melting and cooling of chocolate in order to

achieve at the end the correct crystalline structure of the constituent cocoa butter

among the six existing polymorphic forms, namely the form V (Schenk and Peschar,

2004). The control of this process is important for the quality of the product, as well-

tempered chocolate is shiny, even-colored, snap, and smooth tasting, while badly-

tempered chocolate is chewy (not knocking), chalky and grainy, within the form of an

unattractive, dull brown mass because of fat blooming.

As the different polymorphic forms have different melting ranges, the trick to

tempering is to carefully control the temperature of the melted chocolate (Stapley, et

al.,1999). More precisely, the chocolate is first melted, then cooled to initiate

nucleation of seed crystals of the V form, and finally reheated slightly by a few

degrees to melt out crystals of any lower melting polymorphs. However, the

temperature history is not the only key for success. The importance of subjecting the

chocolate to a carefully defined shear has been recently recognized as a key factor in

F. Debaste et al.

the success of tempering processes (Donshi and Stapley, 2006). Indeed, even the

earliest hand tempering methods subjected the molten chocolate to shear by spreading

it across a marble slab with a knife.

Alternative tempering processes consist in inoculating melted chocolate with

crystallization nuclei of form V such as solid chocolate or coca butter in order to

induce the crystallization of the chocolate under this desired form V. Here also the

control of temperature is of high importance to remain with enough stable seed

crystals when the chocolate is finally allowed to fully cool.

The main objective of the present work is to investigate the usefulness of a modern

modelling tool, Computation Fluid Dynamic (CFD), on the study and control of a

given chocolate tempering process, considered here as one reference process of the

food industry.

CFD is indeed a simulation tool for the numerical solution of fluid flow and heat and

mass transfer problems. The transport equations describing the conservation of mass,

momentum and energy are solved numerically to give predictions of, for example,

velocity shear, temperature and pressure profiles inside the studied system. It is

therefore a potentially very interesting tool to study a process such as the tempering

one, where local time evolutions of temperature are notably of high importance.

While CFD is a technique that has a variety of applications in different processing

industries, it is only in recent years that it has been applied to food processing, mainly

for sterilization, mixing and drying processes (Norton and Sun, 2006)

2. Methods and materials

2.1. The tempering process

The tempering process considered in this study is one used by professional pastry

chefs and chocolate makers for dipping, molding and decorating. A batch of chocolate

is first molten in a bowl placed in a temperature controlled water bath. The molten

batch is then cooled at ambient temperature, while gently mixed by hand (Figure 1).

Figure 1 A tempering bowl

Contribution to the modelling of chocolate tempering 3

At a given temperature, the liquid chocolate is seeded with a given amount of small

solid pieces (for instance crystals of cocoa butter) to ensure the crystallization under

the best form (V).

2.2. The research methodology

The main purpose of the present study is to develop a mathematical model to predict

the time evolution of the temperature field inside the chocolate bowl during the

cooling and crystallization periods of the specified tempering process.

Firstly, a mechanical stirrer is designed and manufactured in order to simulate the

manual mixing in a controlled and reproducible manner. A CFD simulation is

performed to characterize the flow in the tempering bowl equipped with this novel

stirrer. These mixing characteristics are studied without taking care of any heat

transport.

Secondly, on the basis of this flow analysis, the transient heat transfer problem is

simplified by neglecting convective terms in the heat balance equation and using an

effective thermal conductivity approach to take into account enhancement of heat

transport by the mixing process. The resulting transient heat balance equation

complemented with adequate boundary conditions is solved using the Comsol

software.

Thirdly, a sink term is added to the heat balance equation to take into account the

additional cooling arising from the latent heat of melting of the solid pieces used as

crystallization seeds. This term is written under the form of a kinetic reaction whose

parameters are identified from Differential Scanning Calorimetry (DSC) as well as

from melting experiments carried out under adiabatic conditions.

Finally, the predicted transient temperature profiles are validated against tempering

experimental results obtained in the laboratory, using the bowl of Figure 1 and the

new stirring device.

2.3. The experimental and measuring equipments

A scheme of the batch tempering bowl used in the laboratory is given in Figure 2. It is

axisymmetrical, shaped as about an half sphere, with a depth at the centre of 7 cm and

a diameter at the free surface of 20 cm. Three such bowls were at our disposal for our

tempering experiments: one made out of stainless steel, one made out of glass, and the

last one made out of plastic. Those three materials, which are all currently used by

chocolate makers, have quite different thermal properties (conductivity and specific

heat).

The location of the 9 thermocouples is given in Figure 2 (indicated as T0 to T8).

These are connected to a data acquisition system which records the temperature every

second.

F. Debaste et al.

The bowl is equipped with a novel stirrer (see Figure 3) specially designed to

reproduce in the melted chocolate motions similar to those induced by hand mixing. It

consists of two tilt incurved blades fit out with rabbets at specific radial locations in

order to enable the rotation of the stirrer in presence of the thermocouples set

vertically in the bowl. During the melting of the chocolate, the bowl is immersed in a

tank filled up with thermoregulated water. After seeding of solid particles at the

chosen temperature, the cooling is operated either in the tank whose control

temperature is switched down to the desired temperature either by leaving the bowl at

ambient air. All the experimental results presented here are obtained using dark

chocolate produced by Belcolade manufacturer and using cocoa particles as solid

seeds. Their properties are given in Table 1.

Figure 2 : Scheme of the tempering bowl (vertical diametral plane) and location of 9 thermocouples T0 to T8

Figure 3 : New stirrer

k (W/m K) Cp (J/kg K) (kg/m3)

Dark chocolate (liquid) 0,16 – 0,26 1670 (40 – 60 °C) 1260 (35 °C) Cocoa butter (solid) 0,09 (21°C) 2010 (15 – 21 °C) 960 - 990 (15°C) Cocoa butter (liquid) 0,12 (43°C) 2090 (32 – 82 °C) 880 - 900

Table 1 : Properties of the products used in the experiments


Besides, DSC measurements, using a DSC 821e from Mettler-Toledo, are operated to

investigate the kinetics of the melting of the solid seeds.

In addition, an adiabatic tank of 1 liter, equipped with thermocouples and a mixing

device, is used to measure the latent heat of melting of those particles. This latent heat

can indeed be calculated thanks to a suitable transient heat balance fed with the

recorded temperature fall induced by the melting of the particles.

2.4. The CFD softwares

Computational Fluid Dynamics are performed using Fluent software code, well-

known to tackle problems from most major industry sectors. It is based on a finite

volume method and is used here to solve Navier-Stokes equation in the stirred bowl.

The simplified transient heat balance is solved using the commercial code Comsol

Multiphysics, based on finite elements technique.

3. Results and analyses

3.1. Design of the stirrer

A continuous mixing is needed during the whole tempering process, in order to

maintain the solid seeds in uniform suspension and to avoid inner temperature

gradients during the cooling of the melted chocolate thanks to increased heat

transfers.

These objectives can easily be reached by gentle hand mixing. However, in this study,

a mechanical device is preferred in order to ensure reproducible and constant mixing

conditions.

It is first observed that an uniform distribution of the solid seeds can not be reached

using standard stirrers such as anchorages or impellers; indeed, most of the seeds are

left floating on the surface which leads to the formation of an upper solid layer when

the particles melt as well to the apparition of large temperature fluctuations. A new

stirrer is then designed, which apparently reproduces the expected mixing effects

during a tempering experiment.

CFD simulations are run to study the main characteristics of the flow induced by the

stirrer. Even if it can be anticipated that melted chocolate exhibits non-Newtonian

behavior because of the importance of solid particles (of sugar, cocoa and milk

powder) dispersed in the continuous fat melted phase, a laminar Newtonian flow is

assumed.

Indeed, as can be seen on Figure 4 where experimental viscosities are plotted as a

function of shear rates and fitted using an Ostwald-De Waele model, the viscosity

remains nearly constant for shear rates higher than 2 s-1. CFD simulations carried out

using Fluent software show that, for the rotation speed used for tempering

experiments (32 rpm), shear rates lie within this range where the viscosity can be

assumed to be nearly constant, around 40.000 cp (Figure 5). Therefore, it can be

expected that CFD simulations performed using a Newtonian model in laminar

F. Debaste et al.

regime will enable to get at least a good qualitative description of the main features of

the flow. Theses ones are given in Figure 6.

It can be observed that the stirrer produces an important axial downward motion in

the centre of the bowl. which should lead to an efficient distribution of the solid seeds

within the mass of melted chocolate and enable to avoid temperature fluctuations as

explained above. This is indeed verified experimentally.

Figure 4 : Viscosity as a function of shear rates

Figure 5 : Computed shear rates (rotating speed of 32 rpm)

Figure 6: (left) vertical downward components of reduced velocity in two horizontal planes (at 2 and 6 cm from

the bottom of the bowl); (right) vertical downward component of reduced velocity in a median vertical plane;


3.2. Typical tempering experimental result

A typical tempering experiment is run as follows. Firstly, 2 kg of chocolate is melted

in the bowl at a temperature around 50°C. The stirrer is set in motion at a rotating

speed around 30 rpm, which corresponds to hand agitation. The solid seeds particles

are then introduced into the mass of melted chocolate and dispersed within the

volume thanks to the motions induced by the stirrer device. The wall of the bowl is

cooled down, by contact to ambient air or immersed in a thermo-regulated water bath.

The time evolution of temperature at three locations within the bowl is given for

example in Figure 7; the numbering of the thermocouples is given in Figure 2. Time 0

corresponds to the introduction of the seeds particles. The rapid cooling observed just

after this introduction is mainly due to the endothermic melting of the particles. The

temperature appears to be very homogeneous within the bowl.

The quality of the product obtained by this tempering process will depend on several

operating parameters, such as the ratio between the mass of the particles and the mass

of melted chocolate, the size of the particles, their initial temperature, the temperature

of the melted chocolate at time 0, the temperature of ambient air, the temperature of

the walls during cooling… . Indeed, it is essential that the seed particles provide beta

seeds (form V) for proper tempering; this aptitude will depend somehow or other on

the listed parameters. The development of a mathematical model predicting the

evolution of the temperature in the bowl as a function of those parameters will help in

identifying criteria for good tempering.

Figure 7 : Time evolution of temperature in the bowl during a tempering experiment

3.3. Mathematical Model

The objective of the model under development in this study is to predict the time

evolution of the temperature within the melted chocolate as a function of the

operating parameters mentioned above.

This requires in principle to solve a set of coupled non linear partial derivatives

equations: the balances equations of energy, of continuity and of momentum (Bird et

al., 2002).

F. Debaste et al.

It is proposed here to simplify this problem by introducing an effective thermal

conductivity approach to model the contribution of convection to heat transfer.

Accordingly, the energy equation writes:

QT λt

TρC effp +∆=

∂∂

(1)

where λeff is a lumped thermal conductivity parameter. The enhancement of heat

transport by the mixing process is here modeled as a thermal diffusion process

between neighboring fluid layers. Therefore the continuity and momentum equations

do not have to be solved anymore.

As a consequence of the symmetry of the bowl, temperature T is only function of

radial position r, vertical position z and time t.

Q represents a sink term added to the thermal balance equation to take into account

the additional cooling arising from the latent heat of melting of the solid particles

used as crystallization seeds (Franke, 1998; Norton and Sun, 2006)

This heat balance equation supplemented with a set of appropriate initial and

boundary conditions is numerically solved using Comsol.

As initial condition, assumption is made that at time t=0, temperature is uniform

within the whole bowl and given equal to

0)0( TtT == (2)

The heat flux boundary condition is given at the wall of the bowl and on the free

surface as a sum of two contributions: a convective heat flux and a radiative heat flux:

( ) ( )44

∞−+−=∂∂− TTTThx

Tk sext

xL

σε (3)

where emissivity ε depends on the nature of the emitting surface.

As can be seen in Figure 8, where the ratio of radiative flux to convective flux is

plotted as a function of the temperature, assuming a black body emitting in ambient

air at 20°C, the radiative flux can indeed not be neglected within the temperature

range of interest.

Figure 8 : Comparison of radiative heat flux to convectif heat flux as a function of temperature in the case of a

black body in ambient air at 20°C


3.4. Identification of the parameters of the model

In order to solve numerically the set of equations (1) to (3), values must be given to the parameters of the model: the heat transfer coefficients in the heat flux boundary conditions, the effective thermal conductivity λeff, and the sink term Q.

3.4.1. Heat transfer coefficients

Convective heat transfer coefficients depend on the nature of the fluid, on its velocity,

and on the geometry of the considered system.

The heat transfer coefficient at the top free surface of the bowl is estimated using the

correlation (4), valid for a horizontal free surface subjected to free convection heat

transfer (Holman, 1981): 4/154.0 RaNu = (pour 710<Ra ) (4)

For stirred vessels, the resistance to heat transfer is known to be localized on the inner

wall. It is proposed to use the following correlation developed for anchor stirrers in

laminar regime (Zlokarnik, 1969):

( ) 3/23/1PrRe BCNu += (5)

Anchor can indeed be considered as the standard stirrer whose geometry is the closest

to the one of our new stirrer device.

The values of the heat transfer coefficient on the free surface (hext) and on the inner

wall (hint) calculated using those correlations are given in Table 2. These values are validated against experimental results obtained as follows.

Chocolate is first melted at 50,4°C and than cooled by immersing the bowl in a

thermoregulated water bath set to 32°C. No seeds particles are added and no stirring

is operated. Ambient temperature is measured equal to 24°C.

On the recorded temperatures (Figure 9), it can be observed that there are no

significant differences of temperature between the water and the wall of the bowl

(thermocouples 7, 8 and 10); this shows a very good heat transfer between water and

the bowl and leads us to fix the temperature of the bowl (32°C in this experiment) as

the temperature of the wall.

Besides, due to the absence of any stirring, significant temperature gradients can be

observed within the bowl.

Re Pr C B hext (W/m² K) hint (W/m² K) 97 15.900 0,93 - 4.000 6,8 125

Table 2 : Heat transfer coefficients

F. Debaste et al.

Figure 9 : (a) Position of thermocouples. (b) temperatures as a function of time for cooling without seed particles

Data needed to model this experiment are given in Table 3. The model is run giving the value 0 to the sink term Q and 10,7 W/m K to the

effective thermal conductivity λeff.

The experimental results are compared to the numerical ones in Figure 10.

Considering the very good agreement between them, it can be concluded that the

value of 6,8 W/m² K estimated for the heat transfer coefficient at the free surface, hext,

is adequate.

ρ (kg/m

3)

Cp

(J/Kg K)

Boundary

condition

chocolate/water

Boundary

condition

chocolate/air

Initial

condition

H

(cm)

Time of

simulation

(s)

1290 1650 T = 32 °C hext=6,8

W/m²K

Tair=22,5°C

Twater=32°

Tchoc=50,5°C

6,8 7200

Table 3: Properties of melted chocolate and boundary conditions


Figure 10 : Experimental and numerical temperature profiles during cooling. Initial temperature of chocolate equal

to 50,4 °C. Bowl immersed in water set to 32°C. No seed particles. No stirring. Position of thermocouples is given

in Figure 9.

A second set of experiments is carried out. All the experimental conditions are kept

unchanged except that the stirrer is switched from 0 rpm to 32 rpm. This experiment

is modeled using the values given in Table 2 for the heat transfer coefficients. The

excellent agreement that can be observed in Figure 11 between the experimental and

numerical profile validates the value given to hint.

F. Debaste et al.

Figure 11: Experimental and numerical temperature profile during cooling. Initial temperature of chocolate equal

to 50,4 °C. Bowl immersed in water set to 32°C. No seed particles. Stirring at 32 rpm.

3.4.2. Effective thermal conductivity The value of the effective thermal conductivity keff is identified by least square

optimization using similar experiments carried out without any seeding but keeping

the speed of rotation of the stirrer set to 32 rpm.

It is found that a value of 10,7 W/m K allows to get the best possible comparison

between the numerical and the experimental time evolution of temperature at the

center of the bowl. This value is nearly two orders of magnitude higher than the

thermal conductivity of molten chocolate.

Besides, it is checked that with this value the whole temperature field is predicted

with a very good agreement, whatever the other experimental conditions (mass of

chocolate, initial and boundary temperatures); however, temperature gradients within

the bowl are very low when stirring is operated.

3.4.3. Melting kinetics Melting of seed particles is taken into account in the model using a heat sink term Q.

If we neglect heat needed to bring seeds to their melting temperature and to the

melted chocolate temperature, this term can be expressed as the product of a melting

kinetic rate (1/s) by the latent heat of melting (J/kg)

The kinetics of melting of seeds (particles of form V coca butter) is investigated by

DSC with heating rates between 1°C/min and 20°C/min. One should note that the

term ‘‘kinetics’’ is used in a broad sense in thermal analysis, it indeed covers the

study and modeling of the rate(s) of changes of the measured quantities (Varhegyi,

2007).

For those increasing heating rates, it can be observed that the temperature

corresponding to maximal melting rates varies between 33°C and 40°C (Figure 12).

Those DSC results can be supplemented with the plot of the evolution as a function of

temperature, of the degree of conversion α, namely. the mass fraction of liquid seeds,

calculated thanks to an elementary mass balance (Figure 12).


Figure 12 : DSC results (melting of cocoa butter particles)

It is proposed in this work to describe the kinetic model of melting that occurs under

nonisothermal conditions by equation (6) (Chen et al, 2007):

( ) ( )na

TTR

EA

dt

d αα −

−−= 1exp

0

0 (6)

in which parameters A0, Ea and n can be fitted against DSC results by a least squares

technique. The fitted values are given in Table 4.

Figure 13 gives a comparison of the DSC experimental results to the ones calculated

using expression (6) and parameters given in Table 4. A very good agreement can be

observed.

This validated kinetics expression must be multiplied by the latent heat of melting in

order to end with the sink term Q.

This latent heat is measured using an adiabatic vessel containing melted chocolate and

equipped with a stirring device to maintain a uniform temperature within the

chocolate. Solid seeds are introduced in the vessel and the fall of temperature

resulting from their melting is recorded with type K thermocouples.

As shown in equation (7), latent heat of melting can be calculated using initial

temperatures of seeds and chocolate, their masses (r=M/Mchoc)and specific heat (Cchoc,

Cseeds), and temperature after melting of seeds (Kumano et al., 2007):

Log(A0) (/s) Ea (cal/mol) n

5,34 1001,4 2,8

Table 4: Kinetics parameter fitted to DSC experimental results

Figure 13: Evolution of the mass fraction of liquid chocolate as a function of temperature for three heating rates.

(left) DSC results (right) calculations

F. Debaste et al.

Figure 14: Time evolution of temperature in an adiabatic vessel containing melted chocolate at 61,2 °C. Solid

particles are introduced at 24,5 °C

( )T

r

rCCH seedschoc

f ∆+=∆ (7)

The heat needed to warm up the solid particles to their melting temperature is here neglected. A typical experimental result is presented in Figure 14. The temperature taken to calculate ∆T is the minimum one, reached after about 2 minutes. The next increase in temperature is due to heat dissipation induced by the stirring device. An average value of 180 KJ/Kg is calculated for the latent heat of melting from 4 different experiments, as shown in table 5.

3.5. Experimental validation of the model

The model developed here above is numerically solved with Comsol software.

Its validation is operated following a cross validation approach, i.e. the predicted

results are compared to experimental results that were not used to identify the

parameters of the model.

As an example, the temperature time evolution predicted by the model at the center of

the bowl is compared in Figure 15 to the one measured during a tempering

experiment carried out under the following conditions: Mchoc=2,4 Kg; Mseeds=0,24 Kg;

Tchoc°=51°C; Tseeds°=21°C; Tair=22°C.

The values given to the parameters of the model are the ones presented in the former

subsections of the paper, completed with the law identified for the heat sink term Q:

keff=10,7 W/m K; hext=6,8 W/m² K; hin=125 W/m² K.

As the cooling is operated here by immersing the bowl in a thermoregulated water

bath at 32°C, no radiative transfer has to be taken into account at the walls.

r Tchoc° (°C) Tseeds° (°C) ∆Tf ∆Hf (KJ/Kg) 0,1 61,2 24 10 179,1 0,1 50,1 22 10 180,9

0,1 50 22 10,1 179,1 0,1 60 22 10,2 188,05

Table 5: Values of latent heat of melting deduced from 4 different adiabatic experiments. Cc=1670 J/Kg K;

CB=2010 J/Kg K


Figure 15 : Predicted and measure temperature time evolution during a tempering experiment. Experimental

parameters: Mchoc=2,4 Kg; Mseeds=0,24 Kg; Tchoc°=51°C; Tseeds°=21°C; Tair=22°C. Parameters of the model: keff=10,7 W/m K; hext=6,8 W/m² K; hin=125 W/m² K.

An excellent agreement can be observed between both curves.

Figure 16 presents a similar comparison for other experimental conditions: the initial

temperature of melted chocolate is equal to 44°C and the cooling is operated at

ambient air. In that case, the heat loss by radiation at the wall of the bowl is not

negligible, as can be seen in Figure 16 where the agreement between experiments and

predictions is poor when this radiative heat flux is neglected (grey curve) and very

good when it is taken into account (emissivity ε=0,93 for plastic bowl).

Figure 16: Predicted and measure temperature time evolution during a tempering experiment. Experimental

parameters: Mchoc=2,4 Kg; Mseeds=0,24 Kg; Tchoc°=51°C; Tseeds°=21°C; Tair=22°C. Parameters of the model: keff=10,7 W/m K; hext=6,8 W/m² K; hin=125 W/m² K; ε=0,93

4. Discussion and conclusions

The model developed in this study following the research methodology described in

chapter 2 gives an accurate prediction of the cooling rate and the temperature field

F. Debaste et al.

within a mass of melted chocolate seeded with small solid grains and let at ambient

temperature. As the model uses parameters having a strong physical meaning, it can

now be used to study the effect of different experimental conditions on the cooling

rate of melted chocolate and therefore identify criteria for good tempering.

For instance, the effects of ambient air temperature, of seeds initial temperature and of

mass of chocolate are presented in Figure 17. All three parameters are indeed of

importance for practical applications and end users.

The results of this sensitivity study enable to identify among the different parameters

those which might have a significant influence on the quality of the final product. For

instance, it can be observed that the initial temperature of the seeds should not be a

critical parameter while the ambient temperature should, not surprisingly, have a large

influence in case of cooling at ambient air.

However, in its present state of development, the model is not able to correlate the

prediction of temperature time evolution to the quality of tempering. This requires

indeed additional development of the model as well as experiments.

Besides, on-going works also focus on the development of a shrinking core model to

get a better description of the kinetics of seeds melting (Q).

Finally, the nucleation that takes place later in the second period of the tempering

process will also be studied, in order to complete the model.

Figure 17: Prediction of the influence different parameters on tempering process : (left) ambient temperature

effect (middle) of seeds initial temperature effect (right) the mass of chocolate effect.

5. Nomenclature

A0 pre-exponential factor (s

-1)

C, Cp specific heat (J/kg/K) d characteristic length of a geometry


∆Hf latent heat of melting (J/kg) ∆T difference of temperature (K) Ea activation energy (cal/mol) g acceleration of gravity (m/s²) h convective heat transfer coefficient (W/m² K) hext convective heat transfer coefficient at the free surface (W/m² K) hint convective inner heat transfer coefficient (W/m² K) k thermal conductivity (W/m K) M mass (Kg) Q heat sink term (W/m

3)

R perfect gas constant (cal/mol K) r mass of seeds/mass of melted chocolate T temperature (K) Text ambient temperature (K) T∞t ambient temperature (K) T0 reference temperature (273 K) t time (s) u velocity (m/s) xL direction normal to a surface

Greek symbols α mass fraction of liquid chocolate β coefficient of thermal expansion (K

-1)

εs emissivity of a surface (-) λeff effective thermal diffusion coefficient (m

2/s)

µ dynamic viscosity (Pa s) ν kinematic viscosity (m

2/s)

ρ volumetric mass (kg/m3)

σ Planck constant (W/m² K4)

Subscripts air air water water choc chocolate seeds seeds particles (cocoa butter)

Superscripts ° initial condition Dimensionless numbers Re Reynolds number (du/ν) Pr Prandtl number (Cp µ/k) Ra Rayleigh number (d

3gβ∆T Pr/ν²)

Nu Nusselt number (hd/k)

F. Debaste et al.

6. References,

Bird, R. B., Stewart, W. E., & Lightfoot, E. N. (2002). Transport phenomena. New York: John Wiley & Sons, Inc.

Chen G., LI C., Kuo Y-L.,Yen Y-W (2007) A DSC study on the kinetics of disproportionation reaction

of (hfac)CuI(COD) Thermochimica Acta 456, 89–93

Dhonsi D. & Stapley A.G.F. (2006) The effect of shear rate, temperature, sugar and emulsifier on the tempering of

cocoa butter, Journal of Food Engineering, 77, 936–942

Franke K., Modelling the Cooling Kinetics of Chocolate Coatings with Respect to Final Product Quality. Journal

of Food Engineering, 1998. 36: p. 371-384.

Holman J. P., Heat Transfer. 5th ed. 1981, New York: Mc Graw-Hill.

Kumano H., Asaoka T., Saito A, Okawa S. (2007) Study on latent heat of fusion of ice in aqueous solutions,

International Journal of Refrigeration 30, 267-273

Norton T & Sun D.W. (2006) Computational fluid dynamics (CFD): an effective and efficient design and analysis

tool for the food industry: A review, Trends in Food Science & Technology, 17, 600-620

Schenk H & Peschar R.(2004), Understanding the structure of chocolate, Radiation Physics and Chemistry, 71,

829–835

Stapley, A. G. F., Tewkesbury, H., & Fryer, P. J. (1999). The effects of shear and temperature history on the

crystallisation of chocolate, Journal of American Oil and Chemical Society, 76, 677-685.

Tewkesbury H., Fryer P.J., and Stapley A.G., Modelling temperature distributions in cooling chocolate moulds.

Chemical Engineering Science, 2000. 55: p. 3123-3132.

Varhegyi G (2007) Aims and methods in non-isothermal reaction kinetics J. Anal. Appl. Pyrolysis 79, 278–288

Zlokarnik M (1969), Chem Eng. Tech, 22, 1195

7. Acknowledgements.

F. Debaste acknowledges financial support from the Fonds National de la Recherche

Scientifique, Belgium

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Contribution to the modelling of chocolate tempering process